− = 91 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Let’s try the multiples of 3 as well. Multiples of 2 Multiples of Multiples of 3 Multiples of How Multiples Make Patterns in Numbers Circle the multiples of 2 in the table below. How do the multiples of 2 line up? Let’s check the multiples of other numbers.
92 = × Let’s play “clap number game” by raising hands at the multiples of 2 and clapping at the multiples of 3. 1 Let’s find numbers that are multiples of both 2 and 3. 2 What is the number of the least common multiple of 2 and 3? For 6, raise hands and clap at the same time, right? A number that is a multiple of both 2 and 3 is called a common multiple of 2 and 3. The smallest of all common multiples is called the least common multiple. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 … Multiples of 2 Multiples of 3 Multiples of both 2 and 3 Are there any other numbers which students raise hands and clap at the same time like 6? Activity 1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 6 6 6 7 7 7 7 6 2 4 3 3 Common Multiples
− = 93 Let’s think about how to get the common multiples of 3 and 4. Four friends found different ways to determine the common multiples as follows. Let’s read their ideas and describe each method in sentences. Explain the ideas to your friends. The holes show the multiples. Mero's note multiples of 3 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36… multiples of 4 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 … I fi nd the common numbers from the multiples of 3 and . ThinN about multiples of 3... then, circle the multiples of . 3, 6, 9, 12, 15 18, 21, 24, 27 … Write the multiples of then, circle the multiples of 3. 4, 8, 12, 16, 20, 24, 28, 32, 36 … 3, 6, 9, 12 4, 8, 12 12×2 = 24 , 12×3 = 36 4 Making Tapes of Multiples Place the tape of multiples of 2 on top of the tape of multiples of 3. The common multiples of 2 and 3 are where the holes on both tapes overlap. <amos note 6ares note 9avis note
6 cm 8 cm Coconut Cookies Cookies Coconut Cookies Coconut Cookies Chocolates Chocolates Chocolates Chocolates Coconut Cookies Coconut Cookies Coconut Cookies Coconut Cookies Coconut Cookies PNG Coconut Chocolates Chocolates PNG Cocoa Chocolates 94 = × The least common multiple of 3 and 4 is 12. All common multiples of 3 and 4 are multiples of 12. Write the first 4 common multiples for each of the following groups of numbers. Find the least common multiples. 1 (5, 2) 2 (3, 9) 3 (4, 6) Stack boxes with heights of 6 cm and 9 cm. What is the smallest number where the total heights of the two stacks are equal? Stacked are boxes of cookies with a height of 6 cm each and chocolate boxes with a height of 8 cm each. 1 The total height of the boxes of cookies are multiples of which number? 2 The total height of the chocolate boxes are multiples of which number? 3 What will be the least height that the cookie boxes and chocolate boxes be equal? How many boxes are in each stack? 4 Write the first 3 numbers where the height of both stacks are equal. 5 Exercise 1 2
18 cm 12 cm − = 95 We want to put squares in this frame so there are no gaps. Let’s find out which squares can fit in the frame without any gaps. Think of the length of the sides of the squares when the squares are lined up vertically without any gaps. Place squares of the same size in a 12 cm ×18 cm rectangle. How long is each side of the square? 1 How many cm is each side of the squares when they are lined up vertically over a 12 cm length without any gaps? 1 2 Divisors and Common Divisors Divisor Can we fi t these squares without any gaps? Let’s try to fi t the squares vertically and horizontally.
96 = × 1 2 3 4 6 12 The lengths of the sides of the squares when lined up vertically over a 12 cm length without any gaps are 1 cm, 2 cm, 3 cm, 4 cm, 6 cm and 12 cm. 2 Divide 12 by 1, 2, 3, 4, 6 and 12 one by one to confirm that there are no gaps. Are they divisible by 12? 1, 2, 3, 4, 6, 12 ……Divisors of 12 3 What can you find when divisors of 12 are grouped as shown below? Any number is divisible by 1 and itself. 4 How many cm is each side of the squares when they are lined up horizontally over a 18 cm length without any gaps? The whole numbers by which 12 can be divided with no remainder are called divisors of 12. Think about the length of the sides of the squares when the squares are lined up horizontally without any gaps. 1×12=12 2×6=12 3×4=12 1, 2, 3, 4, 6, 12 ……Divisors of 12 4 cm 4 cm 3 cm 3 cm 2 cm 2 cm 12 cm 1 cm 1cm
− = 97 18 cm 3 cm 3 cm 2 cm 2 cm 1cm 1 cm The lengths of the sides of the squares when lined up horizontally over a 18 cm length without any gaps are 1 cm, 2 cm, 3 cm, 6 cm, 9 cm and 18 cm. 1, 2, 3, 6, 9, 18 ……Divisors of 18 5 How many cm can the sides of the squares be, when lined up vertically and horizontally without any gaps? Height…… 1 2 3 4 6 12 (cm) Width…… 1 2 3 6 9 18 (cm) 6 The common divisors of 12 and 18 are 1, 2, 3 and 6. What is the greatest common divisor of 12 and 18? 18 cm is included because we think only horizontally. The numbers that are divisors of both 12 and 18 are called common divisors of 12 and 18. The largest of all common divisors is called greatest common divisor. We get squares when the width and height are equal. Find all the divisors of 6, 8 and 36 respectively. Write all the common divisors of 8 and 36. 1, 2, 3, 6, 9, 18 ……Divisors of 18 Common Divisors Exercise 1 2
98 = × Let’s think about how to find the common divisors of 18 and 24. Two friends calculated common divisors in different ways in their exercise books but did not complete. Complete their ideas by considering their thinking. Let’s find all the common divisors and then find the greatest common divisors. 1 (8, 16) 2 (15, 20) 3 (12, 42) 4 (13, 9) There are some pairs of numbers like 4 , that have only 1 as a common divisor. We want to divide 8 pencils and 12 exercise books equally amongst the students. What should be the appropriate number of students for distribution? 'ivisors of 1fl 1, 2, 3, , ffi, 1fl 24÷1=24, 24÷2=12, 24÷3=8, 24÷6=4, 24÷9=2 r 6, 24÷18=1 r 6 'ivisors of 1fl, 1, 2, 3, , ffi, 1fl 'ivisors of 2, 1, 2, 3, , , fl, 12, 2 2 3 Exercise 1
− = 99 Let’s think about the divisors of 18. 1 Find the divisors of 18 by arranging 18 square cards to make rectangles. 2 Is 18 a multiple of the divisors you found in 1 ? Some numbers like 2, 3, 5 and 7 are divisible only by 1 and itself. Find such numbers amongst the following numbers. Divide by 2, 3, 4… in order to find them. 9 2 18 • 2 and are divisors of 18. • 18 is a multiple of and 9. 6 3 18 • 3 and 6 are divisors of 18. • 18 is a multiple of 3 and 6. 2 12 22 32 3 13 23 33 4 14 24 34 5 15 25 35 6 16 26 36 7 17 27 37 8 18 28 38 9 19 29 39 10 20 30 40 11 21 31 41 4 The Relationship between Multiples and Divisors Prime Numbers
100 = × A number that can be divided only by one and itself is called a prime number. One is not a prime number. Let’s find divisors of 6. Divisor of 30 is the product of the combination of prime numbers. 2, 3 and 5 are easily found as divisors. 36=6×6 =2×3×2×3 =2×2×3×3 Let’s represent whole numbers by a product form of prime number. 1 Express 6 by a product form of a prime number. 2 Express 30 by a product form of a prime number. 30 = 5 × 6 = 5 × 3 × 2 3 Determine divisors of 30 by using the expression in 2 . Let’s determine the greatest common divisor of 24 and 36 by using a prime number. 24=4×6 =2×2×2×3 When the multiples representations of prime numbers products are compared, it is common to, 2 × 2 × 3 =12. Then, the greatest common divisor is 12. Let’s discuss how to determine the least common multiple of 24 and 36 by using a prime number. 5 6 7 Using Prime Numbers 24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3 Using multiple representation of prime number products, let’s find the numbers that should be multiplied to get the same products? 24× =2×2×2×3× 36× =2×2×3×3×
− = 101 Sieve of Eratosthenes How many prime numbers are there? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Determine a prime number that is less than 100 by the next procedure. 1 Erase 1. 2 Leave 2 and erase multiple of 2. 3 Leave 3 and erase multiple of 3. Leave the first numbers and erase its multiples. Using this method, a prime number like 2, 3, 5, 7, 11, etc, are left. Using this method, find a prime number until 100. Sieve of Eratosthenes is a method that was discovered by a mathematician named Eratosthenes in ancient Greece. He was born in BC (Before Christ) 276 and died BC 194.
102 = × 0, 1, Divide numbers from 0 to 20 into 2 groups by writing them alternately in the two rows below. Start with 0 in the upper row and then 1 in lower row, upper row, lower row, ...sequentially. 1 Divide the numbers in each row by 2. 2 What did you notice when dividing numbers in each row? Arrange the whole numbers into 2 groups as shown below. 3 In which group does 23 belong? How about 98? 4 What rule did you apply when dividing? Identify some situations where we can use even and odd numbers? For the whole numbers, the numbers that can be divided by 2 without remainder are called even numbers and numbers that can be divided by 2 and leaves a remainder 1 are called odd numbers. The flight numbers such as PX240 and PX110 that depart from POM are even numbers. The flight numbers such as PX241 and PX111 that arrive in POM are odd numbers. 0, 18, 36… 176, 212… 1, 19, 37… 177, 213 … How about the scores in sports? 2 1 3 3 Even Numbers and Odd Numbers A B Flight No Time Departure Arrival Time
1 dL 1 dL 1 dL 1 2 (m) 2 0 1 − = 103 Let’s think about numbers up to 50. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 1 Make a list of the multiples of 3. 2 Make a list of the multiples of 7. 3 Make a list of the common multiples of 3 and 7. 4 Make a list of the divisors of 28. 5 Make a list of the divisors of 32. 6 Make a list of the common divisors of 28 and 32. Let’s write the first 3 common multiples of the following pairs of numbers. Then, find the least common multiples. 1 (3, 6) 2 (8, 10) 3 (3, 5) Let’s find all the common divisors of the following pairs of numbers. Then, find the greatest common divisors. 1 (6, 12) 2 (18, 20) 3 (32, 42) 1 2 3 Express the next volume and length by a mixed fraction and an improper fraction. dL dL m m Pages 88 and 98 Pages 92 to 94 Pages 95 to 98 Let’s find all the common divisors of the following pairs of numbers. Do you remember? Grade 4
104 = × Let’s write 3 multiples of the following numbers from the smallest to largest. Find all the divisors for them. 1 16 2 13 3 24 Let’s write 3 common multiples of the following pairs of numbers from the smallest to the largest. Find the least common multiple for them. 1 (3, 7) 2 (12, 18) 3 (10, 20) Let’s write all the common divisors of the following pairs of numbers. Find the highest common divisor for them. 1 (9, 15) 2 (4, 11) 3 (12, 24) PMV bus A departs every 12 minutes and bus B departs every 8 minutes at 4 mile bus stop. Bus A and B both departed at 9 am. What is the next time that bus A and B will depart at the same time? Start with a sheet of graph paper that is 30 cm wide and 12 cm long. Cut out squares of the same size so that no paper is left over. How many cm is each side of the biggest square? How many of these squares can be cut out? Let’s find the prime number that is bigger than 50 and closest to 50. Finding multiples and divisors. Finding common multiples and least common multiples. Finding common divisors and the greatest common divisors. Solving problems by using common multiples or common divisors. Solving problems by using common multiples or common divisors. Understanding some numbers can be divided by only 1 and itself. 1 2 3 4 5 6
− = 105 Topic 2: Traditional Body Counting System Used in Okasapmin Papua New Guinea is home to an extraordinary number of languages and cultural groups. Traditionally, these communities have used diverse and fascinating ways to count and communicate about number. Prof. Geoffery Saxe researched the counting system in Oksapmin, Sandaun province. Let’s find out the counting system of the Oksapmin people. Many groups count by pointing to positions on the body and the Oksapmin people are a good example. As shown in the figure, a person begins on the thumb on one side of the body and counts around the upper body to the little finger on the opposite hand while naming corresponding body parts. To count beyond 27, Oksapmin people continue around the body back up the wrist of the second hand. In traditional life, Oksapmin people used their counting system in several ways. For example, they counted important objects; they indicated order, like points of arrival on a path; they tallied contributions in a bride price exchange. You might be surprised to learn that Oksapmin people did not use their body part counting system to solve arithmetic problems in their traditional activities. However, with the introduction of Australian currency (shillings and pounds) in the 1960’s and in the shift to Papua New Guinea currency (Kina and toea) with independence from Australia in 1975, Oksapmin people developed new ways of using their body system to calculate money when buying and selling goods. Mathematics Practices in Papua New Guinea Source: Professor Geoffery Saxe Ph.D., , University of California, Berkeley, 2013.
106 = × There is 1 2 L of juice in the fraction measuring container. If you draw dividing lines as shown below, how will the quantity be represented? Let’s use fractions to represent the quantity of juice. Let’s pour some orange juice in a fraction measuring container. This is 1 litre container. Can you see scales on the side of the containers. 1 2 1 3 1 5 1 4 L L L 1 5 1 6 1 7 1 4 Fractions 8
− = 107 You can represent the same amount of juice in many different ways in fraction. I can see 1 2 litre represented. I can use 1 2 litre to see the red measurement. L L L 1 6 1 7 1 8 1 9 1 2 1 3 1 8 1 9 1 2
0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 3 1 2 1 4 1 5 1 6 1 7 1 8 1 9 1 0 1 10 1 0 1 11 1 0 1 12 1 0 1 13 1 0 1 14 1 108 = × 1 Let’s explore the equivalence of fractions by using the number line. 1 Equivalent Fractions
− = 109 1 Let’s find fractions, which are equivalent to 1 2 . 1 2 = 4 = 6 = 8 = 5 = 6 = 14 2 Let’s find fractions, which are equivalent to 1 3 . 1 3 = 6 = 3 = 12 3 What numbers are multiplied to each denominator and numerator of the fraction 1 2 in problem 1 ? 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 4 What numbers are multiplied to each denominator and numerator of the fraction 1 3 in problem 2 ? 1 3 = 2 6 = 3 9 = 4 12 Let’s develop 4 fractions which are equivalent to 1 2 . Exercise × × × × × × × × × × × × × × × ×
110 = × = 2 3 4 6 = 6 9 = 8 12 4 6 = 2 3 2× 8÷ 3× 2× 3× 12÷ 8÷ 12÷ 2 4 and 3 4 have same denominator so we can compare them. How can we compare the sizes of 2 3 and 3 4 . The size of fractions does not change even if the numerator and denominator are multiplied or divided by the same number. Let’s think about how to compare 2 3 and 3 4 . 1 Let’s represent 2 3 using various fractions. A Let’s represent 2 3 by 1 6 , 1 9 and 1 12 as the units. B What is the relationship between denominators and numerators of equivalent fractions? Let’s compare the sizes of 2 4 , 2 3 and 3 4 . 1 = = 2 Comparision of Fractions Let’s think about how to compare the size of fractions with different denominators.
Fold into 3 Fold into 4 Fold into 3 Fold into 4 12 1 3 2 4 3 = = − = 111 2 Let’s represent 3 4 by 1 8 , 1 12 and 1 16 as the units. 3 4 = 3× 4× = 12 3 Let’s compare 2 3 and 3 4 by changing their representation using the same denominator. 2 3 = , 3 4 = therefore, 2 3 3 4 Both papers are folded into 12 equal parts. The same fraction can be represented in many ways by changing the units. Let’s Fold a Paper to Compare the Size of Fractions Let’s fold square papers to represent 2 3 and 3 4 as fractions with the same denominator.
112 = × Compare 3 4 and 4 5 by changing them to equivalent fractions with a common denominator. Which denominators can the two fractions below be compared with? Circle them. Finding a common denominator means changing fractions with different denominators into equivalent fractions with the same denominator. Fractions with different denominators can be compared by changing them to fractions with the same denominator. Compare 2 3 and 4 7 by changing them into fractions with common denominators. 2 3 = 21 , 4 7 = 21 , then 2 3 4 7 We can find the common denominator if we multiply denominators of fractions which we would like to compare with. .... 2 3 .... Common Denominators 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 4 5 8 10 12 15 16 20 20 25 24 30 28 35 32 40 36 45 40 50
− = 113 Let’s find the common denominator for 5 6 and 7 8 . Usually, you should choose the least common multiple as the common denominator to use as the smallest common denominator. Let’s compare the following fractions using common denominators. 1 1 4 and 2 7 The least common multiple of 4 and 7 is . 1 4 = 1× 4× = , 2 7 = 2× 7× = , therefore 1 4 2 7 2 1 3 and 2 9 The least common multiple of 3 and 9 is . 1 3 = 1× 3× = , therefore 1 3 2 9 Let’s compare 1 3 4 and 11 6 using a common denominator. I changed improper fraction to mixed fraction. I changed mixed fraction to improper fraction. 4 5 6 Finding Common Denominators Mero’s Idea Yamo’s Idea Choose 24, the least common multiple of 6 and 8, as the common denominator. 5 6 = 5× 6× = 20 24 7 8 = 7× 8× = 21 24 Multiply the two denominators to get the common denominator. 5 6 = 5× 6× = 40 48 7 8 = 7× 8× = 42 48
114 = × Reducing a fraction means dividing the numerator and denominator by a common divisor to make a simpler fraction. Lisa and Joy are looking for fractions that are equivalent to 24 36 and with denominators and numerators smaller than 36 and 24. 1 What rule of fraction are they using? 2 Lisa and Joy got different fractions. Explain their reasons. se ra Ph Lisa Joy 7 Reducing Fractions Because It is a word used to explain, by stating the conclusion first and then explaining why by showing a reason. “ is because ”. For example: We reduce fractions because it makes calculation easier.
− = 115 When we reduce a fraction, we usually divide until we get the smallest numerator and denominator. Steven and Alex reduced 12 18 . Let’s explain their ideas. 1 What are the similarities in their ideas? 2 What are the differences between their ideas? When you reduce a fraction, use the greatest common divisor to reduce the denominator and numerator, just like Alex did in 8 . Let’s reduce these fractions to a common denominator and fill in the with inequality signs. 1 2 3 4 5 2 1 2 3 8 3 5 6 8 9 4 7 12 5 8 Let’s reduce these fractions. 1 8 10 2 3 21 3 16 20 4 18 24 Steven Alex 8 Exercise 1 2
116 = × When we divide 2 L milk amongst students equally, how many litres of milk will each student receive? 2÷ 1 Enter the numbers from 1 to 5 in the and calculate the answers. 2÷ , 2÷ , 2÷ , 2÷ , 2÷ 2 Divide the above expressions into 3 groups based on the answers. A Answers that are whole numbers. ( ) B Answers that are expressed exactly as decimal numbers. ( ) C Answers that are not expressed exactly as decimal numbers. ( ) 2÷3 is 0∙666..., so this cannot be expressed exactly as a decimal number because there is no end. 3 When 2 L is divided equally amongst 3 students. A Colour the part for one student in the diagram. B How many L will each student receive? 1L 1L 1 1 Fractions, Decimals and Whole Numbers Quotients and Fractions Let’s see how to express the quotient of a division problem when it cannot be expressed exactly as a decimal number.
The amount The amount for for one student one student 3 1 1 L L 3 1 L 1 L L 1 L 1 (m) 1 1 2 (m) 2 3 (m) m 1÷4 2÷4 3÷4 m m 0 0 0 4 1 − = 117 The quotient of a division problem in which a whole number is divided by another whole number can be expressed as a fraction. The quotient can be expressed precisely as a fraction. Let’s represent the quotient using a fraction. 1 1÷6 2 5÷8 3 4÷3 4 9÷7 The amount for one student when 1 L is divided into 3 equal parts… L. The amount for one student when 2 L is divided into 3 equal parts… L. 2÷3= Let’s find the length of one section when 1 m, 2 m and 3 m string is divided into 4 equal parts? 1 Let’s write mathematical expressions for 1 m, 2 m and 3 m strings. 2 Let’s find the answers based on a 1 m string? 2 I used 1 3 L from the first 1 L container and 1 3 L from the second 1 L container to fill up the empty container. Exercise
1 1 2 (m) 2 (m) 5 1 0 0.2 0 1 L 1 L 118 = × If we divide a 2 m tape into 5 equal sections, how many metres long will be each section? 1 Let’s express the answer as a fraction and as a decimal number. 2÷5= 2÷5= 2 Let’s write this fraction and decimal number on the number line. Which is larger 3 5 L or 0∙7 L? Let’s express these fractions as decimal numbers or whole numbers. 1 3 10 = 2 29 100 = 3 12 4 =12÷4= 4 1 3 5 = 8 5 =8÷5= 3 5 =3÷5= therefore, 3 5 0∙7 To represent a fraction as a decimal number or whole number, we divide the numerator by the denominator. 3 4 5 Fractions, Decimals and Whole Numbers
Decimal numbers→ Fractions→ 1 1.6 2 1 5 2 5 4 5 1 0 0.6 − = 119 Let’s express 2 and 5 as fractions. 2=2÷1= 2 1 5=5÷1= 2=4÷2= 4 2 5=10÷2= 2=8÷ = 5=30÷ = Let’s express the decimal numbers 0∙19 and 1∙7 as fractions. 1 Since 0∙19 is 19 sets of 0.01, we can think of this as 19 sets of 1 100 and get . 2 Since 1∙7 is sets of 0∙1, we can think of this as 17 sets of and get . Decimal numbers can be expressed as fractions if we choose 1 10 and 1 100 as the units. Whole numbers can be expressed as fractions no matter what number you choose for the denominator. Fill in the with decimals and fractions. 6 7 Exercise
0 1 120 = × Let’s divide the following fractions into 3 groups. 8 10 1 1 2 4 11 3 5 3 1 2 1 3 6 3 A Whole numbers. B Accurate decimal numbers. C Other decimal numbers. Let’s place these numbers on the number line below. 4 11 4 5 0∙6 1 7 20 2 1∙25 1 4 2 3 Changing fractions to decimal numbers makes them easier to compare. 2 3 =2÷3=0∙666…about 0∙67 Whole numbers, decimal numbers and fractions can all be expressed on one number line. That makes it easy to compare numbers. Let’s line up these numbers starting from the smallest. 1∙3 0∙75 4 2 1 1 2 7 10 5 7 Let’s change decimals to fractions and fractions to decimals or whole numbers. 1 0∙9 2 1∙25 3 3 4 4 24 6 5 1 2 5 8 9 Exercise 1 2
0 2 − = 121 Let’s calculate. 1 1 5 + 1 5 2 2 7 + 5 7 3 1 2 4 + 3 4 4 1 5 7 – 6 7 5 2 3 5 –1 4 5 6 2 – 5 8 Let’s change fractions using common denominators by filling in the with inequality signs. 1 2 3 1 2 2 3 4 5 7 3 1 6 5 18 4 6 3 5 12 Let’s reduce these fractions. 1 4 8 2 6 9 3 21 28 4 16 24 5 75 100 Let’s represent their quotients by fractions. 1 1÷7 2 5÷9 3 11÷3 Let’s represent these fractions by decimals or whole numbers. 1 5 10 2 31 100 3 18 6 4 1 1 4 Let’s represent these decimals with fractions. 1 0.3 2 1∙9 3 0∙61 4 1∙11 Let’s write ↓ for numbers on the number line. 1 2 5 0∙7 1 5 20 1∙8 7 5 1 2 3 4 5 6 Pages 110 to 111, 115 Pages 114 to 115 Pages 116 to 117 Pages 118 to 120 Pages 119 to 120 Pages 118 to 120 Do you remember? Grade 4